Short description
(English)
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Uncertainty relations provide fundamental constraints on our ability to do measurements and have been studied from a number of different angles. In particular, one strand of work has studied our ability to predict the outcome when one of two possible measurements is performed on a system, and a second strand has considered the trade-off between how noisy a particular measurement is and how much disturbance it causes. Until now, these two have largely been developed independently. The aim of the present project is to unite the approaches. Specifically, we intend to develop entropic versions of the noise-disturbance uncertainty relation, and further extend these to account for quantum side information. In addition, we aim to develop a generalized uncertainty relation from which the previous relations can be recovered as special cases.
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Abstract
(English)
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Heisenberg's uncertainty principle states that quantum mechanics poses limitations on the joint observation of non-commuting observables. This is usually expressed as the non-existence of quantum states assigning definite values to both observables by lower bounding the associated standard déviations. However, Heisenberg's original formulation is a statement about measurement devices; there is no ideal apparatus, which jointly measures both observables or measures one of them without disturbing the other one. lt is the aim of this project to find new mathematical well-defined notions of Heisenberg's original ideas in terms of operationally accessible quantifies. As reported previously, the main results of this project are in establishing entirely new uncertainty relations based Heisenberg's original ideas - the estimation of errors necessarily imposed on measurement devices which try to simultaneousiy acquire information about two non-commuting measurements. We derived our results both for finite-dimensional Systems as well as for the case of position and momentum measurements. In this previous work, the inability to simultaneousiy détermine the values of two non-commuting variables was expressed either as the impossibility of a joint measurement, or on the disturbance one measurement induces on eigenstates ofthe other one. In the remaining time of this project, we critically reassessed our previous achievements and were able to improve them significantly. We introduced new measures of disturbance, which now describe ail possible variations of Heisenberg's original ideas. Moreover, we derived new uncertainty relations for ail of them, significantly improving our previous results. In addition, we tightened our result on the impossibility of jointly measuring two incompatible observables. Ali results also hold for Heisenberg's original setting of position and momentum measurements. Thus, we achieved the goal to dérive new uncertainty relations expressing the impossibility to simultaneousiy determining the value of two arbitrary quantum observables which complément the aiready existing entropie formulations ofthe uncertainty principle [1]. [1]The uncertainty principle in the présence of quantum memory, M. Berta, M. ChristandI, R. Colbeck, J. M. Renés, R. Renner, Nature Physics, (6) pp. 659, 2010
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